Trig Graphs Tan and Cot

8/2/2019 Trig Graphs Tan and Cot

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Tangent and CotangentGraphs

Reading and Drawing

Tangent and Cotangent Graphs

Some slides in this presentation contain animation. Slides will bemore meaningful if you allow each slide to finish its presentationbefore moving to the next one.


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This is the graph for y = tan x.

This is the graph for y = cot x.

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One definition for tangent is .xcos

xsin

xtan Notice that the denominator is cos x. This indicates arelationship between a tangent graph and a cosine graph.

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This is the graph for y = cos x.


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To see how the cosine and tangent graphs are related, look at whathappens when the graph for y = tan x is superimposed over y = cos x.


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In the diagram below, y = cos x is drawn in gray while y = tan xis drawn in black.

Notice that the tangent graph has horizontal asymptotes(indicated by broken lines) everywhere the cosine graphtouches the x-axis.


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One definition for cotangent is .xsin

xcos

xcot

Notice that the denominator is sin x. This indicates arelationship between a cotangent graph and a sine graph.

This is the graph for y = sin x.

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To see how the sine and cotangent graphs are related, look at whathappens when the graph for y = cot x is superimposed over y = sin x.

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In the diagram below, y = sin x is drawn in gray while y = cot x isdrawn in black.

Notice that the cotangent graph has horizontal asymptotes(indicated by broken lines) everywhere the sine graph touchesthe x-axis.


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y = tan x.

y = cot x.

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For tangent and cotangent graphs, the distance between any twoconsecutive vertical asymptotes represents one complete period.


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y = tan x.

y = cot x.

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One complete period ishighlighted on each of

these graphs.

For both y = tan x and y = cot x, the period is . (From the beginning ofa cycle to the end of that cycle, the distance along the x-axis is .)


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For y = tan x, there is no phase shift.

The y-intercept is located at the point (0,0).

We will call that point, the key point.


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A tangent graph has a phase shift if the key pointis shifted to the left or to the right.


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For y = cot x, there is no phase shift.

Y = cot x has a vertical asymptote located along the y-axis.

We will call that asymptote, the key asymptote.


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A cotangent graph has a phase shift if the keyasymptote is shifted to the left or to the right.


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y = atan b(x - c).

For a tangent graphwhich has no vertical shift,the equation for the graph

can be written as

For a cotangent graphwhich has no vertical shift,the equation for the graph

can be written as

y = acot b(x - c).

c

indicates thephase shift, also

known as thehorizontal shift.

a

indicates whether thegraph reflects about

the x-axis.

b

affects theperiod.


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y = atan b(x - c) y = acot b (x - c)

Unlike sine or cosine graphs, tangent and cotangent graphs haveno maximum or minimum values. Their range is (-, ), soamplitude is not defined.

However, it is important to determine whether ais positive ornegative. When ais negative, the tangent or cotangent graph willflip or reflect about the x-axis.


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Notice the behavior of y = tan x.

Notice what happens to each section of the graph as it nears its asymptotes.

As each section nears the asymptote on its left, the y-values approach - .

As each section nears the asymptote on its right, the y-values approach + .


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Notice what happens to each section of the graph as it nears its asymptotes.

As each section nears the asymptote on its left, the y-values approach + .

As each section nears the asymptote on its right, the y-values approach - .

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Notice the behavior of y = cot x.


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This is the graph for y = tan x.

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y = - tan x

Consider the graph for y = - tan x

In this equation a, the numerical coefficient for the tangent, isequal to -1. The fact that ais negative causes the graph toflip or reflect about the x-axis.


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This is the graph for y = cot x.

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y = - 2cot x

Consider the graph for y = - 2 cot x

In this equation a, the numerical coefficient for the cotangent,is equal to -2. The fact that ais negative causes the graph toflip or reflect about the x-axis.


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y = atanb(x - c) y = acotb(x - c)

baffects the period of the tangent or cotangent graph.

For tangent and cotangent graphs, the period can be determined by

.b

period

Conversely, when you already know the period of a tangent or

cotangent graph, bcan be determined by

.period

b


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A complete period (including two consecutive vertical asymptotes) hasbeen highlighted on the tangent graph below.

The distance between the asymptotes in this graph is .

Therefore, the period of this graph is also .

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For all tangentgraphs, the period isequal to thedistance betweenany two consecutivevertical asymptotes.

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.2

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period

b

We will leta

= 1, butacould be any positive

value since the graph hasnot been reflected aboutthe x-axis.

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2Use , the period of this tangent graph, to calculate b.

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31 ba

An equation for this graph can be written as xy23tan1

or .xy2

3tan


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A complete period (including two consecutive vertical asymptotes) hasbeen highlighted on the cotangent graph below.

The distance between the asymptotes is .

Therefore, the period of this graph is also .

0x 4x

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For all cotangentgraphs, the period isequal to thedistance betweenany two consecutivevertical asymptotes.

4

864202468


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.4

1

4

period

b

We will let a= 1, but acould be any positive

value since the graphhas not been reflectedabout the x-axis.

4Use , the period of this cotangent graph, to calculate b.

4

11 ba

An equation for this graph can be written as

or .

xy 4

1

cot1

864202468

xy4

1cot


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y = tan x has no phase shift.

We designated the y-intercept, located at (0,0),as the key point.


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y = cot x has no phase shift.

We designated the vertical asymptote on the y-axis (at x = 0)as the key asymptote.

x = 0


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If the key point on atangent graph shifts tothe left or to the right,

or if the key asymptote

on a cotangent graphshifts to the left or tothe right,

that horizontal shift iscalled a phase shift.


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y = a tan b (x - c)

cindicates the phase shift of a tangent graph.

For a tangent graph, the x-coordinate of the key point is c.

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For this graph,c

= because the key point shifted spaces to theright.

An equation for this graph can be written as .

2

2

2

tan xy


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y = a cot b (xc)

cindicates the phase shift of a cotangent graph.

For a cotangent graph, cis the value of x in the key vertical asymptote.

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For this graph, c= because the key asymptote shifted left to .

An equation for this graph can be written as or

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cot xy

.2

cot xy


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Graphs whose equations can be written as a tangent function can alsobe written as a cotangent function.

Given the graph above, it is possible to write an equation for thegraph. We will look at how to write both a tangent equation thatdescribes this graph and a cotangent equation that describes thegraph.

The tangent equation will be written as y = atan b(xc).

The cotangent equation will be written as y = acot b(xc).

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For the tangent function, the values for a, b, and cmust be determined.

This tangent graph has reflected about the x-axis, so amust benegative. We will use a = -1.

The period of the graph is .

The key point did not shift, so the phase shift is 0. c= 0

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periodb


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041 cbaThe tangent equation for this graph can be written

as or .)0(4tan1 xy xy 4tan


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For the cotangent function, the values for a, b, and cmust bedetermined.This cotangent graph has not reflected about the x-axis, so amust

be positive. We will use a = 1.

The period of the graph is .

The key asymptote has shifted spaces to the right , so the

phase shift is . Therefore, .

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period

b

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cbaThe cotangent equation for this graph can be written

as .

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4cot xy


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It is important to be able to draw a tangent graph when you are giventhe corresponding equation. Consider the equation

Begin by looking at a, b, and c.

.6

3tan3

2

xy

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2 cba


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.6

3tan3

2

xy

The negative sign here means that the tangent graph reflects or flips

about the x-axis. The graph will look like this.

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2a


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.6

3tan3

2

xy

b = 3

3

b

period

Use bto calculate the period. Remember that the period is the distance

between vertical asymptotes.


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.6

3tan3

2

xy

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cThis phase shift means the key point has shifted spaces

to the right. Its x-coordinate is . Also, notice that the keypoint is an x-intercept.

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The period is ; half of the period is . Therefore, the

distance between the x-intercept and the asymptotes on either side is .

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3tan3

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xy

Since the key point, an x-intercept, is exactly halfway between two verticalasymptotes, the distance from this x-intercept to the vertical asymptote oneither side is equal to half of the period.

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.6

3tan3

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xy

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We can use half of the period to figure out the labels for vertical

asymptotes and x-intercepts on the graph. Since we already

determined that there is an x-intercept at , we can add half of the

period to find the vertical asymptote to the right of this x-intercept.6

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x-interceptHalf of the period

Verticalasymptote


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.6

3tan3

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xy

Continue to add or subtract half of the period, , to determine the

labels for additional x-intercepts and vertical asymptotes.6

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VerticalasymptoteHalf of the period

x-intercept

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It is important to be able to draw a cotangent graph when you aregiven the corresponding equation. Consider the equation

Begin by looking at a, b, and c.

.8

x4cot3y

8c4b3a


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The positive sign here means that the cotangent graph does not reflect

or flip about the x-axis. The graph will look like this.

3a .

8x4cot3y


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b = 4

4b

period

Use bto calculate the period. Remember that the period is the distance

between vertical asymptotes.

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x4cot3y


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8c

This phase shift means the key asymptote has shifted

spaces to the left. The equation for this key asymptote is

.

8

8x

08

8

x4cot3y


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The period is ; half of the period is . Therefore, the

distance between asymptotes and their adjacent x-intercepts is . Thisinformation can be used to label asymptotes and x-intercepts.

The distance from an asymptote to the x-intercepts on either side of it isequal to half of the period.

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x4cot3y

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Sometimes a tangent or cotangent graph may be shifted up or down. Thisis called a vertical shift.

y = atan b(x - c) +d.

The equation for a tangent graph with a vertical shift can be written as

The equation for a cotangent graph with a vertical shift can be written as

y = acot b(x - c) +d.

In both of these equations, drepresents the vertical shift.


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A good strategy for graphing a tangent or cotangent function that has avertical shift:

Graph the function without the vertical shift

Shift the graph up or down dunits.

Consider the graph for .

The equation is in the form whered equals

3, so the vertical shift is 3.

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x4cot3y

dcxbcotay

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x4cot3yThe graph of was drawn in the previous example.

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To draw , begin with the graph for .

Draw a new horizontal axis at y = 3.

Then shift the graph up 3units.3

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The graph now represents .

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x4cot3y

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x4cot3y

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x4cot3y

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x4cot3y

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This concludes

Tangent and CotangentGraphs.

Trig Graphs Tan and Cot

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Transcript of Trig Graphs Tan and Cot